On the spectral theory of Gesztesy-Seba realizations of 1-D Dirac operators with point interactions on a discrete set

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:254

Article

Carlone, Raffaele^1,2  Malamud, Mark³  Posilicano, Andrea⁴ 

[1] Univ Naples Federico II, Dipartimento Fis, Naples, Italy

[2] Ist Nazl Fis Nucl, Sez Napoli, Naples, Italy

[3] NAS, Inst Appl Math & Mech, Donetsk, Ukraine

[4] Univ Insubria, DiSAT Sez Matemat, Como, Italy

关键词: Dirac operators;    Local interactions;    Non-relativistic limit;    Self-adjointness;    Discreteness;    Continuous spectrum;   

DOI  :  10.1016/j.jde.2013.01.026

来源: Elsevier

【 摘 要 】

We investigate spectral properties of Gesztesy-Seba realizations D-x,D-alpha and D-x,D-beta of the 1-D Dirac differential expression D with point interactions on a discrete set X = {x(n)}(n=1)(infinity) subset of R. Here alpha := {alpha(n)}(n=1)(infinity) and beta := {beta(n)}(n=1)(infinity) subset of R. The Gesztesy-Seba realizations D-x,D-alpha and D-x,D-beta are the relativistic counterparts of the corresponding Schrodinger operators H-x,H-alpha and H-x,H-beta with delta- and delta'-interactions, respectively. We define the minimal operator D-x as the direct sum of the minimal Dirac operators on the intervals (x(n-1), x(n)). Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator D-x* in the case d(*)(X) := inf{vertical bar x(i) - x(j)vertical bar, i not equal j} = 0. It turns out that the boundary operators B-x,B-alpha and B-x,B-beta parameterizing the realizations D-x,D-alpha and D-x,D-beta are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrodinger operators with point interactions. We show that certain spectral properties of the operators D-x,D-alpha and D-x,D-beta correlate with the corresponding spectral properties of the Jacobi matrices B-x,B-alpha and B-x,B-beta, respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy-Seba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light c -> infinity. Most of our results are new even in the case d(*)(X) > 0. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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